Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 7 / Solution

For a nonzero solution, the separable differential equation gives $dy/y=3x^{2}dx$. Integrating yields $\ln |y|=x^3+C$, hence $y=K{e}^{x^3}$. The initial condition fixes $K=3/e$, soThe zero solution omitted by division by $y$ cannot satisfy the given initial condition. Differentiating the displayed solution verifies $y^{\prime }=3x^{2}y$.